Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $x = \dfrac{p - 9}{p^2 + 6p - 27} \times \dfrac{2p^2 + 14p - 60}{-4p - 40} $
Answer: First factor out any common factors. $x = \dfrac{p - 9}{p^2 + 6p - 27} \times \dfrac{2(p^2 + 7p - 30)}{-4(p + 10)} $ Then factor the quadratic expressions. $x = \dfrac {p - 9} {(p - 3)(p + 9)} \times \dfrac {2(p - 3)(p + 10)} {-4(p + 10)} $ Then multiply the two numerators and multiply the two denominators. $x = \dfrac {(p - 9) \times 2(p - 3)(p + 10) } { (p - 3)(p + 9) \times -4(p + 10)} $ $x = \dfrac {2(p - 3)(p + 10)(p - 9)} {-4(p - 3)(p + 9)(p + 10)} $ Notice that $(p - 3)$ and $(p + 10)$ appear in both the numerator and denominator so we can cancel them. $x = \dfrac {2\cancel{(p - 3)}(p + 10)(p - 9)} {-4\cancel{(p - 3)}(p + 9)(p + 10)} $ We are dividing by $p - 3$ , so $p - 3 \neq 0$ Therefore, $p \neq 3$ $x = \dfrac {2\cancel{(p - 3)}\cancel{(p + 10)}(p - 9)} {-4\cancel{(p - 3)}(p + 9)\cancel{(p + 10)}} $ We are dividing by $p + 10$ , so $p + 10 \neq 0$ Therefore, $p \neq -10$ $x = \dfrac {2(p - 9)} {-4(p + 9)} $ $ x = \dfrac{-(p - 9)}{2(p + 9)}; p \neq 3; p \neq -10 $